Explain the derivation of the formula V = |a.(b x c)| (the volume of a
parallelepiped is equal to the magnitude of the scalar triple product
of the vectors that determine the parallelepiped; where a, b, and c
are those vectors).

The volume of a tetrahedron is one-third the distance from a vertex to
the opposite face, times the area of that face. Find a formula for the
volume of a tetrahedron in terms of the coordinates of its vertices P, Q,
R, and S.

Why is a transpose of a matrix needed? I know how to transpose a
matrix, I just don't know what purpose it serves. I have read through
my text as well as consulted online help without finding any answers.